Newcomb's paradox

One of the most simply stated but astonishing of the so-called prediction
paradoxes that bear on the problem of free
will. It was devised in 1960 by William Newcomb, a theoretical physicist
at the Lawrence Livermore Laboratory and the great-grandson of the brother
of the astronomer Simon Newcomb, while contemplating
the prisoner's dilemma.

A superior being, with super-predictive powers that have never been known
to fail, has put $1,000 in box A and either nothing or $1 million in box
B. The being presents you with a choice: (1) open box B only, or (2) open
both box A and B. The being has put money in box B only if it predicted
you will choose option (1). The being put nothing in box B if it predicted
you will do anything other than choose option (1) (including choosing option
(2), flipping a coin, etc.). The question is, what should you do to maximize
your winnings? You might argue that since your choice now can't alter the
contents of the boxes you may as well open them both and take whatever's
there. This seems reasonable until you bear in mind that the being has never
been known to predict wrongly. In other words, in some peculiar way, your
mental state is highly correlated with contents of the box: your choice
is linked to the probability that there is money in box B. These arguments
and many others have been put forward in favor of either choice. The fact
is there is no known "right" answer, despite the concerted attentions of
many philosophers and mathematicians over several decades.